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Refractive Intervention  |   September 2022
Theoretical Relationship Among Effective Lens Position, Predicted Refraction, and Corneal and Intraocular Lens Power in a Pseudophakic Eye Model
Author Affiliations & Notes
  • Damien Gatinel
    Anterior Segment and Refractive Surgery Department, Rothschild Foundation Hospital, Paris, France
  • Guillaume Debellemanière
    Anterior Segment and Refractive Surgery Department, Rothschild Foundation Hospital, Paris, France
  • Alain Saad
    Anterior Segment and Refractive Surgery Department, Rothschild Foundation Hospital, Paris, France
  • Radhika Rampat
    Anterior Segment and Refractive Surgery Department, Rothschild Foundation Hospital, Paris, France
  • Correspondence: Damien Gatinel, Anterior Segment and Refractive Surgery Department, Rothschild Foundation Hospital, 25 Rue Manin, 75019 Paris, France. e-mail: gatinel@gmail.com 
Translational Vision Science & Technology September 2022, Vol.11, 5. doi:https://doi.org/10.1167/tvst.11.9.5
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      Damien Gatinel, Guillaume Debellemanière, Alain Saad, Radhika Rampat; Theoretical Relationship Among Effective Lens Position, Predicted Refraction, and Corneal and Intraocular Lens Power in a Pseudophakic Eye Model. Trans. Vis. Sci. Tech. 2022;11(9):5. doi: https://doi.org/10.1167/tvst.11.9.5.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

Purpose: To ascertain the theoretical impact of anatomical variations in the effective lens position (ELP) of the intraocular lens (IOL) in a thick lens eye model. The impact of optimization of IOL power formulas based on a single lens constant was also simulated.

Methods: A schematic eye model was designed and manipulated to reflect changes in the ELP while keeping the optical design of the IOL unchanged. Corresponding relationships among variations in ELP, postoperative spherical equivalent refraction, and required IOL power adjustment to attain target refractions were computed for differing corneal powers (38 diopters [D], 43 D, and 48 D) with IOL power ranging from 1 to 35 D.

Results: The change in ELP required to compensate for various systematic biases increased dramatically with low-power IOLs (less than 10 D) and was proportional to the magnitude of the change in refraction. The theoretical impact of the variation in ELP on postoperative refraction was nonlinear and highly dependent on the optical power of the IOL. The concomitant variations in IOL power and refraction at the spectacle plane, induced by varying the ELP, were linearly related. The influence of the corneal power was minimal.

Conclusions: The consequences of variations in the lens constant mainly concern eyes receiving high-power IOLs. The compensation of a systematic bias by a constant increment of the ELP may induce a nonsystematic modification of the predicted IOL power, according to the biometric characteristics of the eyes studied.

Translational Relevance: Optimizing IOL power formulas by altering the ELP may induce nonsystematic modification of the predicted IOL power.

Introduction
The purpose of intraocular lens (IOL) power calculation formulas is to determine the refractive power of an implant that will allow the operated eye to achieve the target refraction.1 The formula precision and accuracy depend on the reliability of the preoperative biometric measurements, the difference between the achieved versus predicted effective lens position (ELP), and possible postoperative fluctuations in corneal power.24 The quality of the postoperative ELP prediction is the most critical factor in controlling residual refractive error.5 The ELP does not necessarily coincide with the physical position of the IOL front surface, equatorial plane, or back surface. In theoretical thick lens paraxial ocular models, the ELP corresponds to the distance separating the principal image plane of the cornea and the principal object plane of the IOL.6 Manufacturers label IOL powers with consistent values, but details of the IOL itself, such as geometrical and optical design, are not as readily available. For the same anatomical position, two implants of the same labeled power that are located at the same physical distance from the corneal endothelium will induce different refractive outcomes if their optical designs are not identical. This variation corresponds to differences in position of the principal object plane of any given IOL depending on its design. The lens constant compensates for these and many other parameters, such as minor systematic differences in power values that arise from the test methods.6 
In most cases, optimizing an IOL power formula involves canceling a systematic bias—that is, the non-null arithmetic mean of the prediction error (PE), calculated as the difference between the measured and predicted postoperative spherical equivalent (SE). This process, sometimes referred to as zeroization, involves selecting constant values that would adjust for a null mean arithmetic prediction error.7 Such customization cannot be generalized and is only valid for a dedicated environment (e.g., for one surgical center with standardized surgical techniques and measurement equipment). Hence, a change of lens constant is generally necessary to optimize a formula to a different type of implant than that for which it has been originally adjusted. When more clinical data are available for various IOLs, their respective lens constants can be refined. The lens constant correlates the postoperative physical location of the lens component of the IOL to its effect on the final refraction.8,9 In most formulas based on an optical model, a constant adjustment of the lens equates to modifying the predicted position of the IOL by a specific increment.10 In this context, the modification of the constant is similar to an incremental change in the predicted position for all implants. In the case of a paraxial optical model in thick lenses, the modification of the lens constant is equivalent to the displacement of the principal object plane of the IOLs. 
Several methods exist for this task, depending on the context and structure of the formulas. As part of the comparison of IOL power calculation formulas, through prediction of the postoperative SE, it is possible to communicate the value of the arithmetic mean of the postoperative refractions of the holdout dataset. This dataset contains the preoperative biometric parameters of interest, as well as the type and power of each implanted IOL. The lens constant is then adjusted for each IOL type within each of the compared formulas, until the arithmetic mean of the predicted refractions is equal to the mean value communicated.7 This trial-and-error constant optimization can also be performed for the eyes of a smaller external dataset of randomly chosen patients from a non-biased cohort. 
When optimizing formulas from a fully populated dataset of operated eyes, formulas with one lens constant can be reorganized and solved for that constant. For each documented clinical case, one can back-calculate which lens constant is required to yield the refraction achieved after cataract surgery. The mean or median of all individual constants is selected as an optimized constant for the dataset.11 
Whatever the context and the strategy used, derivation of customized IOL constants for formulas such as the SRK/T, Hoffer Q, Holladay 1, and single-optimized Haigis formulas results in the determination of one lens constant, which will be added to a function that predicts the ELP. Whether optimizing a formula on a large sample of eyes that have already been operated on or on a hidden dataset, each of the IOL powers used in the baseline computations may influence the constant (value of the theoretical displacement necessary to compensate for the systematic bias in a given eye). If the initially calculated IOL power is retained, moving each IOL by the same increment compared to their baseline predicted ELP would induce a refraction difference. Therefore, when used on a new dataset with the adjusted lens constant, the considered formula will predict a different IOL power to achieve the desired postoperative refraction. Hence, the value of the final increment in ELP allowing the zeroization of the PE (i.e., the lens constant adjustment) will depend on the distribution of the IOL powers within the sample or dataset used for the baseline power calculation. 
In this article, we used a thick lens eye model to explore the theoretical relationships between the variations in thick lens effective lens position (ELPT) and predicted postoperative refraction for various IOL and corneal powers. A variation in ELPT may be induced by a variation in optical design and/or physical displacement of the implant. When the ELPT of each IOL is changed by the same increment, without any change in the design of its optics or its haptics, this is equivalent to an anatomical displacement of each IOL. We first investigated the theoretical variation in ELPT needed to compensate for a given systematic bias (non-null arithmetic mean of PE) as a function of the respective IOL and corneal powers of the considered eye. We then explored the relationships between incremental changes in ELPT and the SE prediction in spectacle plane corrections for various IOL and corneal powers. Finally, we calculated the difference in power applied to a given IOL to compensate for the change in spectacle refraction induced by a variation of its effective position. 
Material and Methods
Thick Lens Schematic Pseudophakic Eye Model
Paraxial optics formulas for calculating the respective optical powers and principal plane positions of the cornea and IOL, modeled as thick lenses, and the resultant power and principal plane positions have been reviewed in a previous publication.12 Herein, we describe an explicit formula allowing back-calculation of the theoretical position of the principal object plane of an IOL. In what follows, the formulas are clearly reported for a pseudophakic eye modeled as thick lenses with four refractive surfaces and distinct refractive indices between the aqueous and vitreous humor (Fig. 1). 
Figure 1.
 
Paraxial, four-surface, thick-lens, pseudophakic eye model. Fe is the back focal point of the paraxial schematic pseudophakic eye. ALA is the anatomic axial length from the anterior surface of the cornea to the photoreceptor plane (PP). It is equal to the distance S1Fe if the eye is emmetropic. The ALT is the thick lens axial length, which connects the principal image plane of the cornea Hc to the PP and is reduced by the distance HiHi separating the two principal planes of the IOL. ALP is the anterior lens position connecting the corneal vertex (S1) to the IOL vertex (S3); ELPT is the effective thick lens position joining the principal image plane of the cornea to the principal object plane of the IOL; and d is the distance to the spectacle plane.
Figure 1.
 
Paraxial, four-surface, thick-lens, pseudophakic eye model. Fe is the back focal point of the paraxial schematic pseudophakic eye. ALA is the anatomic axial length from the anterior surface of the cornea to the photoreceptor plane (PP). It is equal to the distance S1Fe if the eye is emmetropic. The ALT is the thick lens axial length, which connects the principal image plane of the cornea Hc to the PP and is reduced by the distance HiHi separating the two principal planes of the IOL. ALP is the anterior lens position connecting the corneal vertex (S1) to the IOL vertex (S3); ELPT is the effective thick lens position joining the principal image plane of the cornea to the principal object plane of the IOL; and d is the distance to the spectacle plane.
The cornea is comparable to a convex–concave lens where the refractive index is attributable to its main layer, the corneal stroma (ns). The total corneal power is denoted Dc. It can be obtained from the value of the anterior and posterior radii of curvature (Rca and Rcp) and the refractive indices of air, stroma, and aqueous humor (na). 
The optical power of a thick intraocular lens is denoted Di. It depends on the curvatures of its anterior and posterior surfaces (Ria and Rip), their separating distance (corresponding with the central IOL thickness, di), and the index variations between that of the media in contact with these surfaces and that of the lens itself. In this work, the power of the IOL was calculated from identical refractive indices of the aqueous humor and vitreous (n = 1.336). 
In a thick lens model, depicting the distance between the refractive elements included in the computations involves the position of the principal planes of these elements. The effective thick lens position (ELPT) corresponds to the distance between the position of the principal image plane of the cornea and the principal object plane of the IOL. It can be computed from the corneal and IOL design characteristics and the anterior lens position (ALP), which separates the respective anterior surfaces of the cornea and the IOL. The axial length of the thick lens eye model (ALT) differs from the anatomical axial length (ALA). It is computed as the difference between the principal corneal image plane and the entire eye focal image point reduced by the distance between the principal planes of the IOL. However, the derived formulas apply to a thin lens eye model, where the axial length and the effective position of the implant have a more direct correlation. 
For a given corneal power Dc and IOL power Di, the distance separating the principal image plane of the cornea (Hc) from the focal image point of the entire eye (Fe) is given by12 
\begin{equation}H{{\rm{^{\prime}}}_c}F{{\rm{^{\prime}}}_e} = EL{P_T} + \frac{{{n_v} - \frac{{{n_v}}}{{{n_a}}}{D_c}EL{P_T}}}{{{D_c} + {D_i} - \frac{{{D_c}{D_i}EL{P_T}\ }}{{{n_a}}}}}\end{equation}
(1)
 
In an emmetropic eye, HcFe = ALT, where ALT is the axial length of the thick lens model eye. 
Determination of the Incremental Change in ELPT Required to Compensate for a Systematic Bias
From the schematic eye, one can calculate the refraction (SE = R) at the spectacle plane (d) using the vergence formula:  
\begin{equation}R = - \frac{1}{{\frac{1}{{\frac{{{n_a}}}{{\frac{{{n_a}}}{{ - \frac{{{n_v}}}{{\left( {A{L_T} - EL{P_T}} \right)}} + {D_i}}} - EL{P_T}}} + {D_c}}} - d}}\end{equation}
(2)
 
If all the parameters concerning the design of the IOL are considered fixed, a variation of the ELPT is equivalent to an anatomical displacement of the implant. 
Solving Equation 2 for ELPT allows us to compute the variation in the effective thick lens position (ΔELPT) that is necessary to compensate for a change in spectacle refraction (ΔR). Within the context of a zeroing procedure, this corresponds to a systematic bias (non-null mean PE) requiring compensation:  
\begin{equation}\Delta EL{P_T} = \frac{{ - B + \sqrt {{B^2} - 4AC} }}{{2A}} - \ EL{P_T}\end{equation}
(3)
where  
\begin{eqnarray*} A &=& {D_i}\left( {d\left( {R + \Delta R} \right){D_c} + {D_c} - \left( {R + \Delta R} \right)} \right)\\ B &=& - \ {n_a}\big( d\left( {R + \Delta R} \right)\left( {{D_c} + {D_i}} \right) - {D_c} - {D_i} \\ &&- \left( {R + \Delta R} \right) \big) - \left( {{D_i}A{L_T} - {n_v}} \right)\\ &&\times\left( {d\left( {R + \Delta R} \right){D_c} - \left( {R + \Delta R} \right) - {D_c}} \right)\\ C &=& \ {n_a}\big( \big( A{L_T}\left( {R + \Delta R} \right)\left( {d\left( {{D_c} + {D_i}} \right) - 1} \right) \\ &&- d\left( {R + \Delta R} \right){n_v} - A{L_T}\left( {{D_c} + {D_i}} \right) + {n_v} \big) \big) \end{eqnarray*}
 
Theoretical Impact of the Increment in ELPT on the Refraction of the Pseudophakic Eye
Equation 3 is derived from Equation 2 and makes it possible to directly calculate the theoretical impact of a variation of the ELPT on the refraction of the pseudophakic eye according to the power of the implant and the corneal diopter. This equation enables computation of the impact on R of a given variation in ELPTELPT), analogous to a change in the lens constant:  
\begin{eqnarray}{\rm{\Delta }}R = - \frac{1}{{\frac{1}{{\frac{{{n_a}}}{{\frac{{{n_a}}}{{ - \frac{{{n_v}}}{{\left( {A{L_T} - \left( {EL{P_T} + {\rm{\Delta }}EL{P_T}} \right)} \right)}} + {D_i}}} - \left( {EL{P_T} + {\rm{\Delta }}EL{P_T}} \right)}} + {D_c}}} - d}} - R\nonumber\\ \end{eqnarray}
(4)
 
We limited our simulations to eyes having received positive power implants (Di > 0) between 1 D and 35 D. 
Table 1 displays the values used for the numeric simulations. All modeled eyes were emmetropic for their initial ELPT, corneal and IOL powers, and axial length. 
Table 1.
 
Numeric Values Used to Design Baseline Emmetropic Pseudophakic Eyes
Table 1.
 
Numeric Values Used to Design Baseline Emmetropic Pseudophakic Eyes
The refractive index values used for numerical simulations were na = 1.336 (aqueous), nv = 1.336 (vitreous), nc = 1.376 (corneal stroma), and ni = 1.45 (IOL material). All modeled IOL had a symmetrical shape (null Coddington shape factor). The central thickness of the cornea was tc = S1S2 = 0.535 mm. The selected anterior and posterior corneal radii of curvature were Rca = 8.7 mm and Rcp = 7.5 mm (Dc = 38 diopters [D]); Rca = 7.7 mm and Rcp = 6.8 mm (Dc = 43 D); and Rca = 6.9 mm and Rcp = 6 mm (Dc = 48 D). The distance to the spectacle plane was set to d = 12 mm. 
Relationship Between the Change in Refraction (ΔR) Versus the Change in IOL Power (ΔDi) Induced by a Specified Change in ELP (ΔELPT)
Let Di be the power of an IOL located at the effective lens position (ELPT). The change in IOL power (ΔDi) required to keep the target refraction constant when an amount of ΔELPT shifts the IOL position is given by  
\begin{eqnarray}\Delta {D_i} &=& \frac{{{n_v}}}{{\left( {\ H{{\rm{^{\prime}}}_c}F{{\rm{^{\prime}}}_e}\ - \ (EL{P_T} + \Delta EL{P_T})} \right)}} \nonumber\\ &&- \frac{{{n_v}}}{{\left( {\ H{{\rm{^{\prime}}}_c}F{\rm{^{\prime}}}{{\rm{^{\prime}}}_e}\ - \ (EL{P_T} + \Delta EL{P_T})} \right)}}\end{eqnarray}
(5)
where HcF′′e corresponds to the distance separating the principal image plane of the cornea from the focal point of the entire eye when the effective lens position is at ELPT + ∆ELPT (Fig. 2) and can be computed using Equation 1
Figure 2.
 
Moving the implant increment equal to ΔELPT results in a displacement of the focal point of the entire eye. The vergence of the implant must be adjusted by ΔDi to keep the focal length unchanged.
Figure 2.
 
Moving the implant increment equal to ΔELPT results in a displacement of the focal point of the entire eye. The vergence of the implant must be adjusted by ΔDi to keep the focal length unchanged.
Using Equations 3 and 4, one can investigate the relationship between the variation in refraction (ΔR) versus the required variation in IOL power (∆Di) induced by ΔELPT for various combinations of corneal and IOL powers. 
We considered 12 theoretical emmetropic pseudophakic eyes with various IOL powers (5 D, 15 D, 25 D, and 35 D) and corneal powers (38 D, 43 D, and 48 D). For each eye, we varied the ELPT between −0.17 mm and +0.17 mm by a ±0.1-mm increment and computed (1) the resulting change in refraction in the spectacle plane (ΔR) and (2) the necessary change in IOL power (ΔDi) to maintain emmetropia. For each combination of IOL and corneal power, linear regression was performed between the corresponding values of ΔR and ΔDi to obtain the regression coefficient. 
Results
Determination of the Incremental Change in ELPT Required to Compensate for a Systematic Bias
The required change in ELPTELPT) to compensate for various systematic bias values (analogous to variations in refraction, ΔR) from −0.3 D to +0.3 D by 0.1-D steps were computed for different IOL powers (Di) ranging from 1 D to 35 D (1-D steps) and three different total corneal powers (Dc): 38 D, 43 D, and 48 D (Figs. 3a,  2b, 3c, respectively). 
Figure 3.
 
Theoretical change in ELPTELPT) necessary to induce a given variation in the spectacle plane refraction (ΔR) as a function of IOL power (Di) for total corneal power: (a) Dc = 38 D, (b) Dc = 43 D, and (c) Dc = 48 D. A positive change corresponds to an increase in the ELPT.
Figure 3.
 
Theoretical change in ELPTELPT) necessary to induce a given variation in the spectacle plane refraction (ΔR) as a function of IOL power (Di) for total corneal power: (a) Dc = 38 D, (b) Dc = 43 D, and (c) Dc = 48 D. A positive change corresponds to an increase in the ELPT.
ΔELPT increased dramatically with low-power IOLs (less than 10 D), proportional to the magnitude of the change in refraction (ΔR). The flatter the cornea, the higher the difference required for the same planned refractive variation, but the incurred impact was low. 
Theoretical Impact of the Increment in ELPT on the Refraction of the Pseudophakic Eye
There was a linear variation of the change in refraction (ΔR) with IOL power, which increased proportionally with ΔELPT (Figs. 4a–4c). A positive variation of the ELPTELPT > 0) resulted in a hyperopic shift in the refraction. 
Figure 4.
 
Change in spectacle plane refraction (ΔR) for various increments in ELPT as a function of IOL power (Di) for total corneal power: (a) Dc = 38 D, (b) Dc = 43 D, and (c) Dc = 48 D.
Figure 4.
 
Change in spectacle plane refraction (ΔR) for various increments in ELPT as a function of IOL power (Di) for total corneal power: (a) Dc = 38 D, (b) Dc = 43 D, and (c) Dc = 48 D.
The higher the corneal power, the more significant the change in predicted SE, but the influence of the corneal power was low. 
Relationship Between the Change in Refraction (ΔR) Versus the Change in IOL Power (∆Di) Induced by ΔELPT
Figure 5 displays the relationship between the change in spectacle refraction and required change in IOL power to maintain emmetropia incurred by a shift of effective lens position (ELPT). The regression coefficient expresses the change in spectacle refraction induced by a variation of 1 D in IOL power (Fig. 5). The regression coefficient values were between 0.617 and 0.731. The values of ΔR and ΔDi obtained for ΔELPT = +0.1 mm and ΔELPT = −0.1 mm are indicated for each simulation. Table 2 displays the values for the 12 scenarios. 
Figure 5.
 
Relationship between the change in spectacle refraction and required change in IOL power to maintain emmetropia incurred by a shift of effective lens position (ELPT) between −1.7 mm and +1.7 mm by 0.1-mm steps for a pseudophakic eye implanted with an IOL of power Di = 25 D and total corneal power Dc = 43 D.
Figure 5.
 
Relationship between the change in spectacle refraction and required change in IOL power to maintain emmetropia incurred by a shift of effective lens position (ELPT) between −1.7 mm and +1.7 mm by 0.1-mm steps for a pseudophakic eye implanted with an IOL of power Di = 25 D and total corneal power Dc = 43 D.
Table 2.
 
Coefficients of Linear Regression (Estimated Ratio ± 1.96 SD) Between IOL Power and Spectacle Refraction for Various Combinations of Corneal and IOL Powers
Table 2.
 
Coefficients of Linear Regression (Estimated Ratio ± 1.96 SD) Between IOL Power and Spectacle Refraction for Various Combinations of Corneal and IOL Powers
Discussion
Determining the optimum value of a lens constant occurs under various circumstances and can be achieved by various methods. For IOL power calculation formulas that use a single lens constant, its value correlates to predicting the ELP. 
Our results were computed using a thick lens eye model and show that, for the same target refractive change (ΔR), the required theoretical variation in ELPT is inversely proportional to the power of the implanted IOL. For implants with a power greater than 8 D, the theoretical variation in the effective lens position necessary to induce a refractive correction of ±0.1 D is less than 250 µm. For IOL powers less than 8 D, this variation increases exponentially: ±1 mm of a shift in ELPT is required to induce a refractive change of ±0.1 D for an IOL power of +2D. This tendency is relatively insensitive to the value of the corneal power. 
Hence, to compensate for the same refractive bias (zeroization), the required theoretical variation in ELPT tends to be less for short eyes with high-power IOLs and increases dramatically for long eyes with low-power IOLs. This suggests that, for optimization methods that would be based on the determination of an average or median of the optimal constants calculated for each eye of a dataset, the determination of the lens constant will be greatly affected by the distribution of IOL powers within the dataset used. It also suggested that eyes with low-power IOLs will have a stronger influence than eyes with high-power IOLs. 
Once determined, the change in the lens constant will result in an identical increment in the predicted effective lens position, which will not affect the predicted refraction of all eyes equally. As shown in Figure 4, the refractive consequences of the same variation in effective IOL position will vary according to the biometry characteristics of the eye concerned. The refraction of eyes with a high-power IOL requirement is much more sensitive to lens constant adjustments than that of eyes with a low-power IOL requirement. 
Figure 4 can be interpreted as the theoretical impact that a single change such as an identical offset between the haptics and the optic for all IOL powers would have on the postoperative refraction, before performing the lens constant adjustment of the considered formula. The magnitude of the predicted change in spectacle refraction would increase exponentially for shorter eyes receiving high-power lenses. This nonlinearity means that the change in spectacle refraction will be strongly influenced by the distribution of implant powers within the database used. We presented the theoretical impact of IOL design variations on postoperative refraction, which was greater for high-power implants.12 This result suggests that the optimization processes induced by a change in implant design (without theoretical variation of their anatomical position) are subject to the same effects and the predominance of high-power implants. Due to manufacturing constraints and the variation in optical thickness related to changes in the refractive index and curvature of IOL surfaces, it is likely that the differences in paraxial refraction induced by different IOLs are related to the conjunction of anatomical displacement of the lens body and a variation in their optical design. 
Holladay et al.13 recently put forward convincing arguments in favor of choosing the standard deviation (SD) of the prediction error as the most relevant criterion for comparing the results of formulas with each other after lens constant adjustment is made to nullify the arithmetic mean PE. However, our results suggest that the lens constant value to cancel the systematic bias is likely to unpredictably vary the SD of the recalculated PE. The data presented in Figure 4 show the influence of the implant power on the variation of theoretical refraction for the same variation of the ELP. On the other hand, Table 2 reveals that the ratio between the variation of IOL power and the variation in refraction is relatively constant throughout the IOL power range. An example of an unfavorable scenario for optimization is a formula whose systematic bias would be mainly linked to a larger PE in long eyes (having received a low-power implant, such as <10 D). Our calculations show that the quality of the ELP prediction is not a major determining criterion for long eyes. Yet, as the zeroization process will eventually shift the predicted effective position for all eyes by a constant increment, it may degrade the performance of the formula on short eyes receiving high-power implants. Our results suggest that taking into account the power distribution of implants could be useful in the context of formula optimization. 
When analyzing the series retrospectively, some investigators have used the intraocular lens prediction error obtained by subtracting the predicted IOL power targeting the actual refraction following cataract surgery from the power of the IOL implanted.14,15 It is often assumed that 1 D of IOL prediction error results in 0.7 D of refractive error at the spectacle plane.1618 However, the theoretical relationships between the refractive and the IOL power errors have not been extensively explored. Our results show that this ratio is valid for most biometric configurations but tends to be slightly different in extreme eyes. A constant increment of the IOL power induces a linear variation of the predicted refraction (Fig. 5), whose coefficient is relatively independent of the power of the considered IOL (Table 2). The compensation of a systematic refractive bias could be achieved by altering the target refraction by an amount equivalent to this bias. This would amount to adding or subtracting some constant value to the nominal power that would have been calculated for the IOLs without this adjustment. Such a zeroization process, based on a systematic variation added to the refractive target, would be less sensitive to the IOL power distribution of the dataset. It remains to be determined if the relative consistency of the ratio between the IOL and refractive prediction error would better preserve the performances obtained by the formula on the considered group before optimization when eliminating a systematic error. This would be achieved by adjusting for each eye (up or down) by an amount proportional to the arithmetic mean PE. Some of the dispersion and average error in refractive accuracy are related to variations in IOL design and anatomical position. These variations occur across powers of the same IOL type. Optimizing a formula by adding an offset to the target refraction would have a more consistent effect over the entire power range of the IOLs compared with altering the predicted ELP. However, a constant increment of the ELP is justified to address the impact of an average change in the expected position on postoperative refraction caused by a change in IOL style. To overcome these problems, it may be beneficial to know the design characteristics of the implants to improve the calculation methods using thick lens or ray-tracing-based IOL power formulas. 
Our results were limited to formulas using a single lens constant for their optimization and do not perfectly apply to formulas with more than one constant, such as the Haigis and Castrop formulas.10,19,20 Instead of the straightforward calculation of the lens constants using formula inversion, nonlinear optimization algorithms have been developed with very high performance that could optimize any target parameter with any optimization criterion. It can be a measure that has high relevance for the patient and the patient's refractive outcome, such as the mean, the mean absolute, the median, or the root mean square error (RMSE) in terms of deviation of the achieved refraction after cataract surgery from the formula predicted refraction.19 Some prerequisites and methods for successful formula constant optimization have been published recently20 as a nonlinear gradient descent method to search for an optimized constant that yields the lowest mean absolute or RMSE. 
Our evaluation was simplified by using theoretical eyes initially focused on an object at infinity, even though in clinical practice an IOL may be chosen so that the best focus of the eye is at a closer distance. 
Conclusions
Our results highlight the influence of the preoperative biometry characteristics of eyes in the lens constant-based optimization process according to the influence of the IOL power. The corresponding increment in ELPT is inversely proportional to the IOL power of the considered eye for the compensated systematic bias. Subsequently, the variation of a lens constant has a greater influence on the power computation in short eyes. This is due to high-power IOLs having a significant impact on the predicted refraction, even for small variations in ELPT. The optimization of a formula based on multiple constants and nonlinear algorithms is certainly more robust against these biases. Nevertheless, any computational process requiring the adjunct of a constant shift in the predicted ELP value will be theoretically subject to these variations. 
Acknowledgments
  
Disclosure: D. Gatinel, None; G. Debellemanière, None; A. Saad, None; R. Rampat, None 
References
Khoramnia R, Auffarth G, Łabuz G, Pettit G, Suryakumar R. Refractive outcomes after cataract surgery. Diagnostics (Basel). 2022; 12(2): 243. [CrossRef] [PubMed]
Norrby S. Sources of error in intraocular lens power calculation. J Cataract Refract Surg. 2008; 34(3): 368–376. [CrossRef] [PubMed]
Kugelberg M, Lundström M. Factors related to the degree of success in achieving target refraction in cataract surgery: Swedish National Cataract Register study. J Cataract Refract Surg. 2008; 34: 1935–1939. [CrossRef] [PubMed]
Hill WE, Abulafia A, Wang L, Koch DD. Pursuing perfection in IOL calculations. II. Measurement foibles: measurement errors, validation criteria, IOL constants, and lane length. J Cataract Refract Surg. 2017; 43(7): 869–870. [CrossRef] [PubMed]
Olsen T. Calculation of intraocular lens power: a review. Acta Ophthalmol Scand. 2007; 85(5): 472–485. [CrossRef] [PubMed]
Cooke DL, Otto B. Lens constants for anterior chamber intraocular lenses. J Cataract Refract Surg. 2021; 47(8): 1094–1095. [CrossRef] [PubMed]
Hoffer KJ, Aramberri J, Haigis W, et al. Protocols for studies of intraocular lens formula accuracy. Am J Ophthalmol. 2015; 160(3): 403–405. [CrossRef] [PubMed]
Olsen T. J Prediction of the effective postoperative (intraocular lens) anterior chamber depth. J Cataract Refract Surg. 2006; 32(3): 419–424. [CrossRef] [PubMed]
Schröder S, Leydolt C, Menapace R, Eppig T, Langenbucher A. Determination of personalized IOL-constants for the Haigis formula under consideration of measurement precision. PLoS One. 2016; 11(7): e0158988. [CrossRef] [PubMed]
Langenbucher A, Szentmáry N, Cayless A, et al. IOL Formula constants: strategies for optimization and defining standards for presenting data. Ophthalmic Res. 2021; 64(6): 1055–1067. [CrossRef] [PubMed]
Wang L, Koch DD, Hill W, Abulafia A. Pursuing perfection in intraocular lens calculations: III. Criteria for analyzing outcomes. J Cataract Refract Surg. 2017; 43(8): 999–1002. [CrossRef] [PubMed]
Gatinel D, Debellemanière G, Saad A, Dubois M, Rampat R. Determining the theoretical effective lens position of thick intraocular lenses for machine learning-based IOL power calculation and simulation. Transl Vis Sci Technol. 2021; 10(4): 27. [CrossRef] [PubMed]
Holladay JT, Wilcox RR, Koch DD, Wang L. Review and recommendations for univariate statistical analysis of spherical equivalent prediction error for IOL power calculations. J Cataract Refract Surg. 2021; 47(1): 65–77. [CrossRef] [PubMed]
Wang L, Booth MA, Koch DD. Comparison of intraocular lens power calculation methods in eyes that have undergone LASIK. Ophthalmology. 2004; 111(10): 1825–1831. [CrossRef] [PubMed]
Francone A, Lemanski N, Charles M, et al. Retrospective comparative analysis of intraocular lens calculation formulas after hyperopic refractive surgery. PLoS One. 2019; 14(11): e0224981. [CrossRef] [PubMed]
Ma JX, Tang M, Wang L, Weikert MP, Huang D, Koch DD. Comparison of newer IOL power calculation methods for eyes with previous radial keratotomy. Invest Ophthalmol Vis Sci. 2016; 57(9): OCT162–OCT168. [CrossRef] [PubMed]
Yang R, Yeh A, George MR, Rahman M, Boerman H, Wang M. Comparison of intraocular lens power calculation methods after myopic laser refractive surgery without previous refractive surgery data. J Cataract Refract Surg. 2013; 39(9): 1327–1335. [CrossRef] [PubMed]
Brenner LF, Gjerdrum B, Aakre BM, Lundmark PO, Nistad K. Presbyopic refractive lens exchange with trifocal intraocular lens implantation after corneal laser vision correction: refractive results and biometry analysis. J Cataract Refract Surg. 2019; 45(10): 1404–1415. [CrossRef] [PubMed]
Langenbucher A, Szentmáry N, Cayless A, et al. Considerations on the Castrop formula for calculation of intraocular lens power. PLoS One. 2021; 16(6): e0252102. [CrossRef] [PubMed]
Langenbucher A, Szentmáry N, Cayless A, Wendelstein J, Hoffmann P. Strategies for formula constant optimisation for intraocular lens power calculation. PLoS One. 2022; 17(5): e0267352. [CrossRef] [PubMed]
Figure 1.
 
Paraxial, four-surface, thick-lens, pseudophakic eye model. Fe is the back focal point of the paraxial schematic pseudophakic eye. ALA is the anatomic axial length from the anterior surface of the cornea to the photoreceptor plane (PP). It is equal to the distance S1Fe if the eye is emmetropic. The ALT is the thick lens axial length, which connects the principal image plane of the cornea Hc to the PP and is reduced by the distance HiHi separating the two principal planes of the IOL. ALP is the anterior lens position connecting the corneal vertex (S1) to the IOL vertex (S3); ELPT is the effective thick lens position joining the principal image plane of the cornea to the principal object plane of the IOL; and d is the distance to the spectacle plane.
Figure 1.
 
Paraxial, four-surface, thick-lens, pseudophakic eye model. Fe is the back focal point of the paraxial schematic pseudophakic eye. ALA is the anatomic axial length from the anterior surface of the cornea to the photoreceptor plane (PP). It is equal to the distance S1Fe if the eye is emmetropic. The ALT is the thick lens axial length, which connects the principal image plane of the cornea Hc to the PP and is reduced by the distance HiHi separating the two principal planes of the IOL. ALP is the anterior lens position connecting the corneal vertex (S1) to the IOL vertex (S3); ELPT is the effective thick lens position joining the principal image plane of the cornea to the principal object plane of the IOL; and d is the distance to the spectacle plane.
Figure 2.
 
Moving the implant increment equal to ΔELPT results in a displacement of the focal point of the entire eye. The vergence of the implant must be adjusted by ΔDi to keep the focal length unchanged.
Figure 2.
 
Moving the implant increment equal to ΔELPT results in a displacement of the focal point of the entire eye. The vergence of the implant must be adjusted by ΔDi to keep the focal length unchanged.
Figure 3.
 
Theoretical change in ELPTELPT) necessary to induce a given variation in the spectacle plane refraction (ΔR) as a function of IOL power (Di) for total corneal power: (a) Dc = 38 D, (b) Dc = 43 D, and (c) Dc = 48 D. A positive change corresponds to an increase in the ELPT.
Figure 3.
 
Theoretical change in ELPTELPT) necessary to induce a given variation in the spectacle plane refraction (ΔR) as a function of IOL power (Di) for total corneal power: (a) Dc = 38 D, (b) Dc = 43 D, and (c) Dc = 48 D. A positive change corresponds to an increase in the ELPT.
Figure 4.
 
Change in spectacle plane refraction (ΔR) for various increments in ELPT as a function of IOL power (Di) for total corneal power: (a) Dc = 38 D, (b) Dc = 43 D, and (c) Dc = 48 D.
Figure 4.
 
Change in spectacle plane refraction (ΔR) for various increments in ELPT as a function of IOL power (Di) for total corneal power: (a) Dc = 38 D, (b) Dc = 43 D, and (c) Dc = 48 D.
Figure 5.
 
Relationship between the change in spectacle refraction and required change in IOL power to maintain emmetropia incurred by a shift of effective lens position (ELPT) between −1.7 mm and +1.7 mm by 0.1-mm steps for a pseudophakic eye implanted with an IOL of power Di = 25 D and total corneal power Dc = 43 D.
Figure 5.
 
Relationship between the change in spectacle refraction and required change in IOL power to maintain emmetropia incurred by a shift of effective lens position (ELPT) between −1.7 mm and +1.7 mm by 0.1-mm steps for a pseudophakic eye implanted with an IOL of power Di = 25 D and total corneal power Dc = 43 D.
Table 1.
 
Numeric Values Used to Design Baseline Emmetropic Pseudophakic Eyes
Table 1.
 
Numeric Values Used to Design Baseline Emmetropic Pseudophakic Eyes
Table 2.
 
Coefficients of Linear Regression (Estimated Ratio ± 1.96 SD) Between IOL Power and Spectacle Refraction for Various Combinations of Corneal and IOL Powers
Table 2.
 
Coefficients of Linear Regression (Estimated Ratio ± 1.96 SD) Between IOL Power and Spectacle Refraction for Various Combinations of Corneal and IOL Powers
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